In this work, we consider the following nonlinear fractional differential equation$$\begin{aligned} {\left\{ \begin{array}{ll} -D^{\nu } u(t)=\lambda f(t,u(t))+e(t) \ \ \ in \ \ (0,1),\\ u^{(j)}(0)=0, \ \ 0\le j \le n-2, \ \ [D^{\alpha } u(t)]_{t=1}=0,\\ \end{array}\right. } \end{aligned}$$where \(\lambda >0\) is a parameter, \(n \ge 3\), \(n-1< \nu < n\), \(1 \le \alpha \le n-2\) and \(D^{\nu }\) stands for the standard Reimann–Liouville derivative, \(f: [0,1]\times [0,+\infty ) \longrightarrow {\mathbb {R}}\) is sign-changing continuous function (that is, we have a so-called equation of semipositone problems). The perturbed term \(e: (0,1) \rightarrow {\mathbb {R}}\) is measurable function and verifies some appropriate conditions. We derive some intervals of \(\lambda \) such that the problem has positive solutions. Our study relies on Guo-Krasnoselskii fixed point theorem.

中文翻译：

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具有扰动项的半正分数阶微分方程正解的存在性。

在这项工作中，我们考虑以下非线性分数阶微分方程$$ \ begin {aligned} {\ left \ {\ begin {array} {ll} -D ^ {\ nu} u（t）= \ lambda f（t， u（t））+ e（t）\ \ \ in \ \（0,1），\\ u ^ {（j）}（0）= 0，\ \ 0 \ le j \ le n-2，\ \ [D ^ {\ alpha} u（t）] _ {t = 1} = 0，\\ \ end {array} \ right。} \ end {aligned} $$其中\（\ lambda> 0 \）是参数，\（n \ ge 3 \），\（n-1 <\ nu <n \），\（1 \ le \ alpha \ le n-2 \）和\（D ^ {\ nu} \）代表标准Reimann–Liouville导数，\（f：[0,1] \ times [0，+ \ infty）\ longrightarrow {\ mathbb {R}} \）是符号改变的连续函数（也就是说，我们有一个所谓的半正问题的方程）。扰动词\（e：（0,1）\ rightarrow {\ mathbb {R}} \）是可度量的函数，并验证一些适当的条件。我们得出\（\ lambda \）的一些间隔，以便问题具有正解。我们的研究依赖于Guo-Krasnoselskii不动点定理。